Lévy-Vasicek Models and theLong-Bond Return Process
Lane P. Hughston
Department of Mathematics
Brunel University London
Uxbridge UB8 3PH, United Kingdom—lane.hughston@brunel.ac.uk

Third London-Paris Bachelier Workshop on Mathematical FinanceParis, 29-30 September 2016—
Joint work with D. C. Brody and D. M. Meier-1-

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This presentation is based on the following:
D. C. Brody, L. P. Hughston & D. M. Meier (2016) Lévy-Vasicek Models andthe Long-Bond Return Process. arXiv:1608.06376L. P. Hughston, September 2016

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Pricing kernels
The Vasicek model is of course one of the oldest and most well studied models
in the mathematical finance literature, and one might think that there is very
little that is new that can be said about it.
But it turns out that there are some surprising features of the Vasicek model
relating to the long rate of interest that are very suggestive when it comes tomodelling long term interest rates in general.
In what follows we shall use a pricing kernel method.
This is not the way in which the Vasicek model is usually presented in theliterature.
But we shall see that the pricing kernel formulation is the most eﬀective for ourpurposes.
Let us recall first how pricing kernels work in the geometric Brownian motion(GBM) model for asset prices.
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We fix a Brownian motion {Wt}t≥0 on (Ω, F, P), and take it to be adapted to afiltration {Ft}.
P is taken to be the real-world measure.
The GBM model is characterised by the specification of a pricing kernel alongwith the price processes of one or more assets.
For the pricing kernel in this model we write
1 2πt = e−rt e−λWt− 2 λ t,(1)

where r is the interest rate, and λ > 0 is a risk aversion parameter.
If an asset pays no dividend over some connected interval of time, we requirethat the product of the pricing kernel πt and the asset price St should be aP-martingale over that interval.

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Third London-Paris Bachelier Workshop-5-If we take this martingale to be of the form
πtSt = S0 e

βWt − 21 β 2 t,

(2)

we obtain
1 2

St = S0 e(r+λσ)t eσWt− 2 σ t,(3)where σ = β + λ is the volatility of the asset.
If the asset has a single payoﬀ HT at time T , and derives its value entirely fromthat payoﬀ, then the value of the asset at time t < T is given by1
Ht = Et[πT HT ].
(4)πt
In particular, if HT = 1, then we recover the pricing formula for a discountbond, given by1(5)PtT = Et[πT ].
πtL. P. Hughston, September 2016

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Third London-Paris Bachelier Workshop-6-Vasicek pricing kernel
To generalize the GBM model, we shall now make the interest rate stochastic
while keeping the risk aversion level constant.
In other words, we consider a pricing kernel of the form
[ ∫ t][]
1πt = exp −rsds exp −λWt − λ2t .
20(6)In the Vasicek model, the short rate process {rt}t≥0 is taken to be amean-reverting Gaussian process of Ornstein-Uhlenbeck (OU) type, satisfying
drt = k(θ − rt)dt − σdWt.

(7)Here k, θ and σ are respectively the mean reversion rate, the mean reversion
level, and the absolute volatility of the short rate.
The dynamical equation (7) can then be solved to give∫ t
rt = θ + (r0 − θ) e−kt − σek(s−t)dWs.

(8)

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To write down the pricing kernel we need the integral of the short rate,∫ trsds.
It =(9)0Substitution of (8) into (9) gives∫ t ∫ s)1(It = θt +1 − e−kt (r0 − θ) − σek(u−s)dWuds.
(10)
ks=0 u=0
The double integral can be rearranged and reduced to give
∫ t()
1σ
It = θt +1 − e−kt (r0 − θ) −(1 − ek(u−t)) dWu.
(11)
kk 0
For some purposes it turns out to be useful to replace the stochastic integral
with an expression involving the short rate to obtain
1σ
It = θt + (r0 − rt) − Wt .
(12)
kk
It follows that the Vasicek pricing kernel can be expressed in the form
[ ()]
()
1 2σ1
πt = exp − θ + λ t +− λ Wt − (r0 − rt) .
(13)
2kk

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Derivation of the discount bond formula
We proceed to work out the price of a discount bond.
Recall thatPtT =1Et[πT ].
πt(14)

Begin by noting that πT is equal to[ ()]∫ T())
1 21(σ k(u−T )σ−kT
exp − θ + λ T − 1 − e(r0 − θ) +−λ− edWu2kkk
0To get an expression for Et[πT ], we need the identity[∫ T (][ ∫ T(]))
2σσ k(u−T )1σσ k(u−T )Et exp−λ− edWu = exp−λ− edu .
kk
2kktt
We then obtain())
1 2
1 ( −kt−kTlog PtT = − θ + λ (T − t) −e −e(r0 − θ)2k∫)∫ tσ k(u−T ))2σ(1 T (σ−λ− edu +1 − ek(t−T )+ek(u−t) dWu.
2 tkkk0L. P. Hughston, September 2016

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Earlier, we showed that∫ t
σek(u−t) dWu = θ + (r0 − θ)e−kt − rt.

(15)

0After some further manipulations one obtains the following expression for thevalue of a T -maturity discount bond:[])()2(211σPtT = exp −R∞(T − t) +.
1 − ek(t−T ) (R∞ − rt) − 3 1 − ek(t−T )k4k(16)Here
λσ 1 σ 2
R∞ = θ +− 2k2k(17)is the asymptotic bond yield, or exponential long rate of interest, defined by
1R∞ = − lim(18)log PtT .
T →∞ T − t

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Uniform integrability of the pricing kernel
We remark on the following feature of the Vasicek model, which doesn’t seem tohave been pointed out before.
The Vasicek model has the property that R∞ > 0 if and only if the pricing
kernel is uniformly integrable.
We recall that a collection C of random variables is said to be uniformlyintegrable (UI) if
lim sup E[|X|1{|X| > δ}] = 0.

(19)

δ→∞ X∈CA random process {Xt}t≥0 is said to be UI if
lim sup E[|Xt| 1{|Xt| > δ}] = 0.

δ→∞(20)t

For a pricing kernel we drop the absolute value sign, and the UI condition islim sup E[πt 1{πt > δ}] = 0.

δ→∞(21)tAn alternative way of expressing this condition is as follows.
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Let us introduce the natural numeraire process {nt}t≥0 (also known as thegrowth optimal portfolio or benchmark portfolio) defined by nt = 1/πt, and setκ = 1/δ.
Then (21) becomes
lim sup E[πt 1{nt < κ}] = 0.

κ→0(22)t

The expression E[πt 1{nt < κ}] is the price at time 0 of a digital put option on
the natural numeraire with strike κ and maturity t.
Therefore the UI property is a condition on a family of option prices:
Proposition 1. A pricing kernel is uniformly integrable if and only if for any
fixed price level ϵ > 0 there exists a strike κ > 0 such that the value of a digitalput option on the natural numeraire is less than ϵ for all maturities.

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Uniform integrability in the Vasicek model
In the case of the Vasicek model, we have the following results.
Proposition 2. If R∞ > 0 then {πt} is uniformly integrable.
This can be proven by applying the Lp-test for uniform integrability.
The L p-test says that if a collection of random variables is bounded in L p forsome p > 1, then it is UI.
The result follows by verifying that there exists a constant γ > 0 and p > 1 suchthat E[πtp] < γ for all t.
Proposition 3. If R∞ < 0 then {πt} is not uniformly integrable.
To see this, recall that if a collection of random variables is UI then it isbounded in L 1.
Thus it suﬃces to show that if R∞ < 0 then {πt} is not bounded in L 1.
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But note that E[πt] = P0t and that[]2(())
1
1σ−kt
−kt 2
P0t = exp −R∞t +1−e
(R∞ − r0) − 3 1 − e.
k4k
We see that if R∞ < 0, this expression grows without bound as t → ∞.

(23)

It remains to check what happens when R∞ = 0.
Proposition 4. If R∞ = 0 then {πt} is not uniformly integrable.
Here, the situation is a little more delicate.
The pricing kernel fails the L p test, so we cannot conclude that it is UI.
But the pricing kernel is bounded in L1, so we cannot conclude that it is not UI.
The conclusion is that we must go to the definition of uniform integrability andsee directly what is going on.

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Recall that uniform integrability is related to the prices of digital puts on thenatural numeraire.
Specifically, we need to investigate limκ→0 supt E[πt 1{nt < κ}].
Following a Black-Scholes type option price calculation, one finds that][1 2At + 2 Bt + log κAt + 12 Bt2
E[πt1(nt < κ)] = eN
Bt
where {At} and {Bt} are deterministic functions given by)(()2)
1(1 σ−kt−λ t +1−eAt = − R∞ +(θ − r0)
2 kk
and( 2)(σ)2() 1 σ2 ()λσσ−kt−2kt
2−λ t−2− 21−e+ 3 1−e.
Bt =kk3k2k
One can show that if R∞ = 0 thenlim sup E[πt 1{nt < κ}] > 0

κ→0

(24)

(25)

(26)

(27)

tand hence the pricing kernel is not UI in this case, as claimed.
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The Lévy–Vasicek pricing kernel
The relation between the uniform integrability of the pricing kernel and thepositivity of the long rate of interest in the Vasicek model leads us to conjecturethat uniform integrability of the pricing kernel may be a condition that we want
to impose on interest rate models in general.
As a step towards understanding what the general situation may be we propose
to investigate a class of Lévy extensions of the Vasicek model.
In particular, building on work by Eberlein & Raible (1999) and Norberg (2004),and others, we use a Lévy process to drive the SDE for the short rate in theVasicek model.
We develop the resulting Lévy–Vasicek models by use of pricing kernel methods.

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We recall that in so-called geometric Lévy models (see, e.g., Brody, Hughston &Mackie 2012) with a constant value for the short rate r, the pricing kernel takes
the formπt = e−rt e−λξt−tψ(−λ).

(28)

Here, {ξt} is a Lévy process, and λ > 0 is a risk aversion parameter.
We assume that {ξt} admits exponential moments.
In particular, we haveE[eαξt ] = eψ(α) t(29)
for α ∈ A for some connected subset A ⊂ R containing the origin, and werequire that −λ ∈ A.
We call ψ(α) the Lévy exponent.

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In the Lévy–Vasicek model, we take the pricing kernel to be of the form][ ∫ tπt = exp −rsds − λξt − ψ(−λ)t ,

(30)

0

where the short rate is assumed to be a Lévy-OU process satisfying a dynamical
equation of the form
drt = k(θ − rt)dt − σ dξt.

(31)

The ensuing calculations follow closely the ones in the classical Vasicek model.
One finds that the pricing kernel in the Lévy-Vasicek model takes the form][)(σ()
1− λ ξt − (r0 − rt) .
πt = exp − θ + ψ(−λ) t +(32)kk
The price of a discount bond is given by[]∫ T()
1PtT = exp − (θ + ψ(−λ)) (T − t) +ψ(αuT ) du +1 − ek(t−T ) (θ − rt) .
kt
Here we have set
σσ k(u−T )αuT = − λ − e.
(33)
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A calculation then shows that the long rate of interest in the Lévy-Vasicek
model is given by(σ)1(34)
R∞ = − limlog PtT = θ + ψ(−λ) − ψ−λ .
T →∞ T − tk
To gain some insight into the significance of this formula, it may be helpful to
re-examine the case when the Lévy process is a Brownian motion.
In that case the Lévy exponent is given by ψ(α) = 12 α2.
Thus we have) 1(σ)2 λσ 1 σ 2
1− λ = λ2 −−λ =− 2.
ψ(−λ) − ψk22 kk2k
Therefore,
λσ 1 σ 2
R∞ = θ +− 2,k2k
in the Brownian case, as we found earlier.
(σ(35)(36)

We see that the two correction terms in the expression for the long rate arise as
the diﬀerence of two Lévy exponents.
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Now we can ask whether in Lévy-Vasicek models positivity of the long rate is
associated with uniform integrability of the pricing kernel, as it is in the classicalVasicek model.
We are able to show the following result:
Proposition 5. The pricing kernel of a Lévy-Vasicek model is UI if and only if
R∞ > 0.
Thus the result we obtain for the Lévy-Vasicek model closely follows that of theclassical Vasicek model.

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References
D. C. Brody, L. P. Hughston & D. M. Meier 2016 Lévy-Vasicek Models and theLong-Bond Return Process. arXiv:1608.06376
O. Vasicek 1977 An Equilibrium Characterization of the Term Structure. Journalof Financial Economics 5, 177–188.
E. Eberlein & S. Raible 1999 Term Structure Models Driven by General LévyProcesses. Mathematical Finance 9, 31–53.
R. Norberg 2004 Vasicek Beyond the Normal. Mathematical Finance 14,585–604.
D. C. Brody, L. P. Hughston & E. Mackie 2012 General Theory of GeometricLévy Models for Dynamic Asset Pricing. Proceedings of the Royal Society ofLondon A 468, 1778–1798.
W. Schoutens 2003 Lévy Processes in Finance Wiley.

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